Luca Vitagliano

The Lagrangian-Hamiltonian formalism for higher order field theories

Abstract We generalize the Lagrangian-Hamiltonian formalism of Skinner and Rusk to higher order field theories on fiber bundles. As a byproduct we solve the long standing problem of defining, in a coordinate free manner, a Hamiltonian formalism for higher order Lagrangian field theories. Namely, our formalism does only depend on the action functional and, therefore, unlike previously proposed ones, is free from any relevant ambiguity.

Secondary calculus and the covariant phase space

Abstract The covariant phase space of a lagrangian field theory is the solution space of the associated Euler–Lagrange equations. It is, in principle, a nice environment for covariant quantization of a lagrangian field theory. Indeed, it is manifestly covariant and possesses a canonical (functional) “presymplectic structure”xa0 ω (as first noticed by Zuckerman in 1986) whose degeneracy (functional) distribution is naturally interpreted as the Lie algebra of gauge transformations. We propose a fully rigorous approach to the covariant phase space in the framework of jet spaces and (A.xa0M.xa0Vinogradov’s) secondary calculus. In particular, we describe the degeneracy distribution of ω . As a byproduct we rederive the existence of a Lie bracket among gauge invariant functions on the covariant phase space.

L∞-ALGEBRAS FROM MULTICONTACT GEOMETRY

Abstract I define higher codimensional versions of contact structures on manifolds as maximally non-integrable distributions. I call them multicontact structures. Cartan distributions on jet spaces provide canonical examples. More generally, I define higher codimensional versions of pre-contact structures as distributions on manifolds whose characteristic symmetries span a constant dimensional distribution. I call them pre-multicontact structures. Every distribution is almost everywhere, locally, a pre-multicontact structure. After showing that the standard symplectization of contact manifolds generalizes naturally to a (pre-)multisymplectization of (pre-)multicontact manifolds, I make use of results by C. Rogers and M. Zambon to associate a canonical L ∞ -algebra to any (pre-)multicontact structure. Such L ∞ -algebra is a multicontact version of the Jacobi bracket on a contact manifold. Unlike the multisymplectic L ∞ -algebra of Rogers and Zambon, the multicontact L ∞ -algebra is always a homological resolution of a Lie algebra. Finally, I describe in local coordinates the L ∞ -algebra associated to the Cartan distribution on jet spaces.

Hamilton–Jacobi diffieties

Abstract Diffieties formalize geometrically the concept of differential equations. We introduce and study Hamilton–Jacobi diffieties. They are finite dimensional subdiffieties of a given diffiety and appear to play a special role in the field theoretic version of the geometric Hamilton–Jacobi theory.

Vector bundle valued differential forms on ℕQ-manifolds

Geometric structures on NQ-manifolds, i.e. non-negatively graded manifolds with an homological vector field, encode non-graded geometric data on Lie algebroids and their higher ana- logues. A particularly relevant class of structures consists of vector bundle valued differential forms. Symplectic forms, contact structures and, more generally, distributions are in this class. I describe vector bundle valued differential forms on non-negatively graded manifolds in terms of non-graded geometric data. Moreover, I use this description to present, in a unified way, novel proofs of known results, and new results about degree one NQ-manifolds equipped with certain geometric structures, namely symplectic structures, contact structures, involutive distributions (already present in liter- ature) and locally conformal symplectic structures, and generic vector bundle valued higher order forms, in particular multisymplectic structures (not yet present in literature).

Generalized contact bundles

Abstract In this Note, we propose a line bundle approach to odd-dimensional analogues of generalized complex structures. This new approach has three main advantages: (1) it encompasses all existing ones; (2) it elucidates the geometric meaning of the integrability condition for generalized contact structures; (3) in light of new results on multiplicative forms and Spencer operators [8] , it allows a simple interpretation of the defining equations of a generalized contact structure in terms of Lie algebroids and Lie groupoids.

Jacobi bundles and the BFV-complex

Abstract We extend the construction of the BFV-complex of a coisotropic submanifold from the Poisson setting to the Jacobi setting. In particular, our construction applies in the contact and l.c.s.xa0settings. The BFV-complex of a coisotropic submanifold S controls the coisotropic deformation problem of S under both Hamiltonian and Jacobi equivalence.

Deformation cohomology of Lie algebroids and Morita equivalence

Let

Deformations of Coisotropic Submanifolds in Abstract Jacobi Manifolds

A Rightarrow M

Deformations of Linear Lie Brackets

be a Lie algebroid. In this short note we prove that a pull-back of